Question Bank by Edumentors
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The radius of a sphere is increasing at a rate of 2 cm/s. Find the rate at which the volume is increasing when the radius is 5 cm.
The volume is increasing at a rate of 628.32 cm³/s when r = 5 cm. Step-by-Step Solution: Volume formula: V = (4/3)πr³ Using chain rule: dV/dt = (4/3)π × 3r² × dr/dt Simplify: dV/dt = 4πr² × dr/dt Substitute r = 5 cm and dr/dt = 2 cm/s: dV/dt = 4π(5)² × 2 = 628.32 cm³/s Key Mistake to Avoid: Students often forge
A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
The probability of drawing two red balls without replacement is 5/8 × 4/7 = 20/56 = 5/14 ≈ 0.357 Step-by-Step Solution: First draw: P(R₁) = 5/8 (5 red out of 8 total) Second draw: P(R₂|R₁) = 4/7 (4 red remaining out of 7 total) Apply multiplication rule for dependent events: P(R₁ and R₂) = P(R₁) × P(R₂|R₁) Calcul
Evaluate ∫ (2x + 1) dx from 0 to 2.
The definite integral ∫ (2x + 1) dx from 0 to 2 equals 6 Step-by-Step Solution: Find antiderivative: ∫(2x + 1)dx = x² + x + C Apply definite integral formula: [x² + x]₀² Substitute upper limit: (4 + 2) Substitute lower limit: (0 + 0) Calculate difference: 6 - 0 = 6 Key Mistake to Avoid: Students often forget to
If 5 painters can paint 5 rooms in 5 days, how long will it take 10 painters to paint 10 rooms?
This problem involves understanding the rate of work. Since 5 painters can paint 5 rooms in 5 days, one painter can paint one room in the same amount of time. This is because the work is directly proportional to the number of workers and the time taken. Therefore, 10 painters can also paint 10 rooms in 5 days, as the
In a triangle, two angles measure 65° and 45°. What is the measure of the third angle, and why?
To solve this, use the fact that the sum of the angles in a triangle is always 180°. If two of the angles are 65° and 45°, then the third angle can be found by subtracting the sum of these two angles from 180°: 180° - (65° + 45°) = 180° - 110° = 70°. So, the measure of the third angle is 70°. This is because the sum
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